Pressure Tendency Equation

 

The Equation of Continuity is

 

 

                                     (1)

 

where  is the three-dimensional wind vector.

 

The hydrostatic equation is

 

 

                                                            (2)

 

Integrate (2) from a level z1 where the pressure is p1=p to the top of the atmosphere at level z2 where p2=0.

 

 

                                                                     (3)

 

Equation (3) states that the pressure at any level is directly proportional to the density of the atmospheric layer of thickness  dz  or, in other words, the weight of the slab of atmosphere of thickness dz.  If level 1 is sealevel and level 2 is the top of the atmosphere, then equation (3) simply states that sealevel pressure is really a function of the  total weight of the atmospheric column.

 

The local pressure tendency can be determined from (3) for a given air column with dz thickness (dz is treated as constant) and is given as:

 

 

                   (4)

 

 

Substitute (1) into (4) to obtain the relationship between pressure tendency at a given level to the mass divergence with respect to and the mass advection  in/out of the air column.

 

                 (5)

 

         Local                   Mass                    Mass

         Pres            due to         In/Out         and    Divergence Or

         Tend                    Advection             Convergence

 

At the synoptic scale, the advection of mass is small and may be dropped from this equation on an order of magnitude basis.  (Note that this term is NOT small near fronts, outflow boundaries, sea-breeze fronts and other mesoscale features and MUST be retained in understanding pressure development with respect to those features.)

 

Thus,  for synoptic scale flow, Equation (5) reduces to

 

 

                                                           (6)

 

 

Substitution of   . into (6) gives

 

 

                              (7)

 

which states that the pressure tendency at the base of an air column is a function of horizontal mass divergence into/out of the air column AND the vertical mass transport through the top and bottom of the air column.

 

The term    represents difference in vertical wind through the air column defined by the depth dz.  As such, the finite differencing yields

 

 

                                  (8)

 

 

If the top of the air column is taken as always at the top of the atmosphere (or troposphere), then w2 = 0 which further simplifies the expression when substituted into (8) and if (8) is substituted into (7).

 

 

                                              (9)

 

 

Equation (9) is the general pressure tendency equation.  This states that  the change in pressure observed at the bottom of a slab of air of thickness dz is due to the net horizontal divergence in/out of the slab modified by the amount of mass being brought in through the bottom of the slab.  In this case, the top of the slab is also the top of the atmosphere. See Fig. 1.

Figure 1

 

To make this equation "relevant" to the surface weather map, remember that at the ground, w1=0.  Hence, the SURFACE PRESSURE TENDENCY EQUATION is:

 

 

                                             (10)

 

where  is the NET HORIZONTAL DIVERGENCE[1] in the air column from the ground to the top of the atmosphere and can be approximated by the product of the NET DIVERGENCE (obtained by summing all the horizontal divergences through the layer) and the thickness of the layer.  Remembering that 90% of the mass of the atmosphere lies beneath the tropopause and that we are treating the density as a mean density for the air column (a constant), then it can be seen that, at a synoptic-scale, surface pressure tendencies are directly related to horizontal divergence patterns in the troposphere. 

 

Equation (10) can be used computationally to obtain qualitative estimates of surface pressure development.  Substitution of equation (2) into the right side of (10) eliminates the density and gravity and allows one to compute the surface pressure tendency on the basis of the net divergence through the layer of dp thickness.

 

Dine’s Compensation

 

The Equation of Continuity is

 

 

                                                                     (1)

 

where  is the three-dimensional wind vector.

 

At the synoptic-scale, gross scaling allows us to drop the local tendency, and the advection of density.  Equation (1) reduces to

 

 

                                                                              (2)

 

which can be rewritten

 

 

                                                                   (3)

 

 

Equation (3) is the so-called “Dine’s Compensation” principle, where the left hand side can be thought of as the “net” horizontal divergence in the layer of ∆z thickness (where the ∆w is estimated from top to bottom of layer), but is really the Equation of Continuity simplified for gross synoptic-scale conditions. 

 

Recall that the Pressure Tendency Equation is

 

 

 

                                                       (4)

 

 

 

If one assumes that the local pressure tendency is negligible (drops on an order of magnitude basis—not true for small time frames), then equation (4) is the same as equation (3).  In Fig. 1, the left hand side of (3) would be the net divergence at A.



[1] Please note that the net divergence is not the mean divergence.  The net divergence represents the arithmetic sum of the divergence from each layer from the bottom to the top of the slab considered.